First in a sequence of two courses designed for beginning graduate students and advanced undergraduates who are interested in solving real-world problems with modern mathematical tools. The sequence is intended to train students who may seek industrial opportunities after graduation. Problems will be approached with a combination of mathematical analysis and scientific computation. The necessary background is elementary differential equations, a working knowledge of computer programming in FORTRAN, Pascal, or the C language, and basic numerical analysis at the level of Mathematics 4640. This course develops the mathematical and computational tools for applications to industrial problems. The course will treat: